Lecture Notes on Conic Sections

Introduction to Conics

• Conics are the next important class of loci after straight lines.
• They have significant applications in calculus and wider mathematics.
• The term "conic" is short for "conic section," which refers to shapes formed by slicing a cone.
• There are four main conic sections:
  • Circle
  • Ellipse
  • Parabola
  • Hyperbola

Geometric Understanding of Conics

• Conics are unified geometrically by their formation from slicing a cone.
  • Circle: A horizontal slice.
  • Ellipse: A slightly tilted slice.
  • Parabola: A slice parallel to the edge of the cone.
  • Hyperbola: A steeper slice than the cone's angle.

Algebraic Representation

• Conics are also unified algebraically:
  • Use of terms like (x^2), (xy), (y^2) alongside constants.
  • Equations involve expressions like (ax^2 + by^2 + cxy).
• Allows for different shaped loci.

Individual Conics

Circle

• Defined as all points equidistant from a central point.
• Equation: (x^2 + y^2 = r^2) where (r) is the radius.

Ellipse

• Similar to a circle but with varying distances from the center.
• Major Components:
  • Semi-major axis (a)
  • Semi-minor axis (b)
• Equation: (\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1)
• The larger distance (a or b) is the semi-major axis.

Parabola

• Graph of a quadratic function.
• Equation: (y = ax^2 + bx + c)
  • Vertex Form: (y = a(x - b)^2 + c).
• The parameter (a) determines width and direction.
  • Positive (a) opens upwards, negative (a) opens downwards.

Hyperbola

• Formed by a steeper slice than the cone's angle.
• Equation: (\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1).
• The arms, as (x) goes to infinity, approach lines with slopes (\frac{a}{b}) and (-\frac{a}{b}).

Applications of Conics

• Astronomy: Historically used to predict planetary orbits.
  • Ancient Greeks assumed circular orbits with epicycles.
  • Kepler proposed elliptical orbits, which is correct for celestial bodies.
• Orbits:
  • Hyperbolic for high-velocity objects (escape velocity).
  • Elliptical for lower velocity, repeating orbits.

Conclusion

• Conics are ubiquitous in mathematics and practical applications, especially in celestial mechanics.